For what a does this system of inequalities have only one solution? { y ≥ x 2 + 4 a 2 x ≥ y 2 + 4 a 2 So, I have tried getting y from the first equation then putting it into second, but then I have two quadratic and x, which of these quadratics are the first term? I know that I have to put it into Δ = 0. But don't really get which of them are first, second etc. Even tried subtracting second to first.

Question

For what a does this system of inequalities have only one solution?      {                            y          ≥                      x            2                    +          4                      a            2                                                x          ≥                      y            2                    +          4                      a            2                                  So, I have tried getting   y from the first equation then putting it into second, but then I have two quadratic and   x, which of these quadratics are the first term? I know that I have to put it into   Δ  =  0. But don&#039;t really get which of them are first, second etc. Even tried subtracting second to first.

Jaelynn Cuevas · Accepted Answer

If   (      x    0    ,      y    0    )  ,      x    0    ≠      y    0   is a solution then   (      y    0    ,      x    0    ) is also a solution. So in order to have only one solution, one must have   x  =  y. Plugging   y  =  x in the first equation one gets:      x    2    −  x  +  4      a    2    ≤  0Which has an unique solution iff   Δ  =  1  −  (  4  a      )    2    =  0 which is equivalent to   a  =  ±      1    4

invioor · Accepted Answer

If   (  x  ,  y  ) is a solution so is   (  y  ,  x  ), so for one solution we need   x  =  y – we thus seek   a for which   y  =      x    2    +  4      a    2   is tangent to   y  =  x, i.e.       x    2    −  x  +  4      a    2   has one solution, hence zero discriminant. Here   Δ  =  1  −  16      a    2    =  0, so   a  =  ±      1    4   where the only solution is   (  x  ,  y  )  =  (  1  /  2  ,  1  /  2  ).

For what a does this system of inequalities have only one solution? { <mtable column

Answered question

Answer & Explanation

New Questions in Algebra I